For thousands of years, angle trisection by means of a ruler and compass has been a problem of interest. In order to realize the significance of this problem, I would like to bring the readers' attention to a few-historical facts that are directly related to it.
The above mentioned problem is so old that it goes back to the ancient times. Even the eminent Archimedes (287-212 B.C.) was trying to solve this problem. Thousands of mathematicians of all generations have thought to solve this problem. Like a magnet it attracts and fascinates the minds of people.
On some occasions, those who have proclaimed to have solved the problem would receive enormous attention, and the issue had been widely advertised in the press. For example in 1931 the president of Duquesne University, Jeremiah Joseph Callahan, declared that he had solved the problem. "TIME" magazine has written an article about the discovery. However, eventually it has been proven that Callahan's solution is wrong.
In 1960, the honorable Daniel Inouye, who represented Hawaii in Congress, had given a speech in which he glorified the achievements of artist M. Kidgell, who according to him was able to solve this problem. M. Kidgell and K. Young has written a book "Two hours, which shook the mathematical world". In 1959 both of the authors traveled all over the U.S. to give lectures about the remarkable achievements of M. Kidgell. San-Francisco TV has even made a special show "Puzzle of Centuries". According to Congressman Inouye the Kidgell solution is taught in hundreds of schools and colleges in the U.S. and also in Canada. The Kidgell solution, however, was also proven to be wrong.
Although, there is already a rigorous proof that this problem cannot be solved with the use of a compass, many people are still trying to solve it, and the French Academy of Science has set a special prize in the case of a solution.
In some other cases, people have proposed new devices that would perform angle trisection. An example is the device of London attorney A. Kempe (shown as FIG. 1A) or another device (FIG. 1B). However, as one may notice, these devices are limited just to the solution of the problem of an angle trisection, and therefore they have only a historical interest.
In general, it must be emphasized that the instruments, which have been invented so far, were proposed and designed in such a way that they would be limited only to perform the solution of the trisection problem or division of an angle into equal parts. Also, they are more complicated than is in fact needed (as will be demonstrated here) to solve the problem of angle trisection.
In view of this, I must immediately indicate that the instrument which is proposed here is not only new in terms of its design and operation to perform the trisection, it is also distinguished from the aforementioned geometric instruments by the fact that its functionality is not limited to the solution of the trisection problem. This new invented instrument may perform all the necessary operations as effectively as the conventional compass.
Let me start with the presentation and solution of a problem that is directly related to this new invention and helps to understand the mathematical background and correctness of the new compass in its operation.
Given: The triangle ABO such, that the side AB=BO, i.e. ABO is an isosceles triangle. PA0 Prove: The angle DAC=.alpha. is a third of the angle DOC=3.alpha.? (see FIG. 2). PA0 Solution: Since ABO is an isosceles triangle, then the angle BAO=BOA=.alpha.(=DAC). Now, since the angle DBO is an exterior angle, then it is equal to the sum of the angles BAO and BOA, i.e. equal to 2.alpha.. By considering the triangle DBO, we can see that it is also. an isosceles triangle, since the sides BO and OD are equal as radiuses. This, in turn, means that the angle ODB is equal to the angle DBO and is equal to 2.alpha.. Finally, the consideration of the triangle DAO and the angle DOC (which is an exterior angle) suggests that DOC is equal to the sum of the angles BAO=.alpha. and ODB=2.alpha.. Thus, DOC=3.alpha.. PA0 Given: An arbitrary angle AOB=3.alpha.. PA0 Find: An angle which is 1/3 of the angle AOB, i.e. angle .alpha. and by that means perform the trisection of the angle AOB? (See FIGS. 3).
This proves the above statement that the angle DAC=.alpha. is a third of the angle DOC=3.alpha.. It also raises an important question, namely, can we construct, by means of a conventional compass, an angle which is a third of an arbitrary given angle and thus to solve the problem of angle trisection.
As a matter of fact, this problem was constructed by Archimedes, when he was considering to solve the trisection problem. He was trying to find, exactly, the point B (or A) such, that after the line was drawn through points B (or A) and D by intersecting the extension of the line OC, the segment AB would be equal to the radius of the circle. As it turned out, it was impossible.
However, as we will learn in the next section, the newly proposed compass easily allows the trisection of an arbitrary angle.